You want to make an old-fashioned pendulum clock, with a period of exactly 1.5 s

Question

You want to make an old-fashioned pendulum clock, with a period of exactly 1.5 second(s). As usual, use the local value of g given in Appendix A.

a) From what length wire should you hang your pendulum bob?

m

b) If you release the bob from an angle of 4.0°, what is the approximate acceleration of the bob at release? (consider sin(?) = ?)

m/s2

c) What is the magnitude of the difference between the actual acceleration and the approximate acceleration? The answer is a very small number. You will need to do the calculation to high precision.

m/s2

d) From this release angle, what is the change in the bob's height from release to the bottom of its arc?

m

e) What is the bob's speed at the bottom of its arc? (use conservation of energy)

m/s

Explanation / Answer

a) Given that, time period of the pendulum, T = 1.5 s angle, = 40 We have, T = 2 *sqrt (L / g) here, L = length of the pendulum           g = acceleration due to gravity = 9.8 m/s2 Therefore, length of the pendulum is given by, L = T2 g / 42 L = (1.5 s)2 (9.8 m/s2 ) / 4(3.14)2 L = 0.56 m _____________________________________________________________________________ _____________________________________________________________________________ b) We have that, acceleration of pendulum, a = - gsin For small oscillations, sin = Therefore, a = - g                       = - (9.8 m/s2 )( 40 )                       = - (9.8 m/s2 ) (4 / 180) rad                        = - 0.6838222 m/s2 Magnitude of acceleration, a = 0.6838222 m/s2 ______________________________________________________________________ ______________________________________________________________________ c) Actual acceleration, a = - g sin                                          = -(9.8 m/s2 ) sin 40                                                                           = -0.6836134 m/s2 Magnitude of actual acceleration, a = 0.6836134 m/s2 Difference between actual acceleration and acceleration is a = 0.6838222 m/s2 - 0.6836134 m/s2 a = 0.0002088 m/s2 ________________________________________________________________________ ________________________________________________________________________ d) Height of the pendulum from its bottom of release is, h = L (1 - cos ) h = (0.56 m) ( 1 - cos 40 ) h = 0.00136 m _______________________________________________________________________ _______________________________________________________________________ e) From the law of conservation of energy ,    Kinetic energy = potential energy           (1/2) mv2 = mgh                           v = sqrt (2gh)                            v = 0.1632 m/s We have, T = 2 *sqrt (L / g) here, L = length of the pendulum           g = acceleration due to gravity = 9.8 m/s2 Therefore, length of the pendulum is given by, L = T2 g / 42 L = (1.5 s)2 (9.8 m/s2 ) / 4(3.14)2 L = 0.56 m _____________________________________________________________________________ _____________________________________________________________________________ b) We have that, acceleration of pendulum, a = - gsin For small oscillations, sin = Therefore, a = - g                       = - (9.8 m/s2 )( 40 )                       = - (9.8 m/s2 ) (4 / 180) rad                        = - 0.6838222 m/s2 Magnitude of acceleration, a = 0.6838222 m/s2 ______________________________________________________________________ ______________________________________________________________________ c) Actual acceleration, a = - g sin                                          = -(9.8 m/s2 ) sin 40                                                                           = -0.6836134 m/s2 Magnitude of actual acceleration, a = 0.6836134 m/s2 Difference between actual acceleration and acceleration is a = 0.6838222 m/s2 - 0.6836134 m/s2 a = 0.0002088 m/s2 ________________________________________________________________________ ________________________________________________________________________ Magnitude of actual acceleration, a = 0.6836134 m/s2 Difference between actual acceleration and acceleration is a = 0.6838222 m/s2 - 0.6836134 m/s2 a = 0.0002088 m/s2 ________________________________________________________________________ ________________________________________________________________________ d) Height of the pendulum from its bottom of release is, h = L (1 - cos ) h = (0.56 m) ( 1 - cos 40 ) h = 0.00136 m _______________________________________________________________________ _______________________________________________________________________ e) From the law of conservation of energy ,    Kinetic energy = potential energy           (1/2) mv2 = mgh                           v = sqrt (2gh)                            v = 0.1632 m/s

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